FNCE 317, Economic Markets
© H Guy Williams, 2006
The Theory of the Firm
Economic Markets
We’ve discussed demand, from the theory of a consumer. For supply we will examine
the firms perspective, what inputs should they use, what are their long run cost functions,
what form do they take? This all leads to aggregate supply, how firms react to different
prices in the market.
We will think of the firm in terms of a production function. If a firm is producing
widgets, to product q widgets it’s going to combine various levels of inputs. We will
keep our analysis simple (as we did in demand), were going to think of production as
coming from two inputs. These inputs can be anything, raw material and labor for
example. But traditionally the way we look at these things is to think of the two inputs as
being labor and capital. But they could be anything and we can generalize this analysis
to many inputs.
Basic Concepts
 The
Production Function
 The firm’s production function for a particular good (q) shows the
maximum amount of the good that can be produced using alternative
combinations of capital (k) and labor (l)
q = f(k, l)
Here we are describing the firm as a production function with variables capital, k, and
labor, l allowing us to produce q goods. We will be interested in understanding how
firms can change the different levels of inputs and what impact that has on the units
produced. We will then think about what the firm is going to maximize. If it wants to
produce so many widgets, what is the optimal level of the inputs, how much capital does
it need and how much labor. Will also ask, if a firm wishes to maximize it’s profits, how
many units should it produce in order to do so. How much does it cost to create these
widgets? Similar to the analysis we did last week but will be some differences.
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Product Curves, cont’d
 Marginal
Product
 To study variation in a single input, we define marginal product as the
additional output that can be produced by employing one more unit of
that input while holding other inputs constant
 For example, the marginal product of capital is given by:
CAPITAL:
MPk 
q
 fk
k
Marginal Product, if we vary one of the inputs, if we increase the amount of labor that we
use, how does it change the number of units that we create? So holding one of the
inputs constant, if we change one of the other inputs how much does it change the
output?
We have a certain fixed level of capital, if I increase my labor hours from 1000 to 1001
how many more units can I product. This is what we are interested in when we are
talking about marginal product. How much more can I produce given one input
increased and the other held constant?
Example, if we focus on capital, the Marginal Product of capital if simply the partial
derivative of the number of units produced, q, wrt the level of capital (we denoting this
as fk). We are looking at the rate of increase of the number of units given an increase in
the level of capital holding labor constant. We derive the labor version of marginal
product by taking the partial derivative wrt labor holding capital constant:
LABOR:
FNCE317
MPl 
Class 8
q
 fl
l
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© H Guy Williams, 2006
Marginal Product, cont’d
 Diminishing
Marginal Productivity
 The marginal product of an input depends on how much of that input is
used
 In general, we assume diminishing marginal productivity
 Using labor as an example, this would mean:
MPl  2 f
 2  fll  0
l
l
We usually assume there is diminishing marginal productivity meaning as we add more
and more of one input while holding the other input fixed we get less bang for the buck.
If I have 50 people working in the factory and only so much capital and I add one more
person I will increase my outputs. As I add the 52 person I’m going to increase the
output but it will do so at a diminishing rate (will not see as must as an increase as we did
when we added number 51). So adding additional inputs holding the other input fixed is
going to increase the output but the rate of change is diminishing, less and less benefit by
adding those factors. This is what is meant by Diminishing Marginal Productivity.
What does this physically mean? Its means that the second partial derivative wrt that
input is negative.
Diminishing Marginal Productivity, cont’d
of diminishing marginal productivity, 19th century
economist Thomas Malthus worried about the effect of
population growth on labor productivity
 But changes in the marginal productivity of labor over time also
depend on changes in other inputs such as capital
 Because
 We need to consider flk which is often > 0
Our model holds the other factors constant, if the other factors can change then it may be
possible to get more productivity out of labor because the derivative will be wrt more
than one term, for instance the partial derivative wrt capital and the partial derivative wrt
labor.
Getting more from the labor we are employing as we add other things including computer
systems and management.
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Product Curves, cont’d
 Average
Product
 The average product of an input, say labor, is the total product
produced divided by the quantity of labor used in its production
q f k, l 
APl  
l
l
 Note that APl also depends on the amount of capital employed
Take a particular number of outputs, say q = 1000 widgets. And I look at the amount of
labor or capital that was necessary to produce that output, the average output is simply
the number of units divided by the inputs employed. We’re looking at the labor example,
could just as easily be capital.
We see that AP is a function of the other input, in this case capital. So dependent on the
amount of capital we have employed, the average product can vary even with constant
levels of labor. If I change the amount of capital, even with the same amount of labor I
can have varying amounts of average product.
Suppose we fix capital at k0. Want to
start with no labor (origin) the increase
the amount of labor (combined with this
fixed level of capital) and we will see an
increase in the level of output. Trying to
q*
average product map the amount of units I’m producing if
I increase labor holding capital fixed,
f(k0, l). Now we will calculate AP at a
l*
l
particular point on the curve by drawing a
line through that point and calculating it’s slope. At this point we are producing q units
with l hours of labor, so the average is just q/l (since the line is through the origin, no
offset).
q
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f(k0, l)
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Output Elasticity
Output Elasticity: if I’m interested in knowing if I can generate a small change in a
quantity that I produce divided by the starting level of production, say I’m at 1000 units
and I make a change, this is going to give a percentage change in the number of units that
I produce, and I want to compare that to a change in one of the input factors that
generates this change in units. Late the second change be labor, if I change the number
of labor hours by a small number divided by the starting level of labor I will get the
percentage change in units produced divided by the percentage change of labor employed
HOLDING CAPITAL K FIXED.
q
q = l q  l F
q l
q l
l
l holding k fixed
where
q F
l l
For example, if I increase my labor hours by 1%, how many percentage point increase do
I get in the number of units that I produce? We could also derive this calculation for
capital holding labor constant.
We take the delta q and l ratios to their limit which gives us the starting labor over the
starting quantity, l/q, multiplied by the partial derivative of the number of units produced
wrt labor. This is the output elasticity of labor. (we can do the same for capital). We
know from the previous slides that
lF =
q l
MPl
APl
This is the ratio of Marginal Product Labor to Average Product Labor.
Output Elasticity is the Marginal Product divided by the Average Product.
When the two are equal a 1% increase in labor will lead approximately to a 1% increase
in output.
q
MPl = Fl
APl =
l
Notice it is possible for MPl < 0 for some levels of l. This would mean we are generating
less output for the one unit of labor added. But this is possible, for instance a factory
running at 100% production fully staffed with workers. If I add more workers to that
factory but don’t actually change the amount of capital assets they operate they actually
end up just getting in each others way. The firm becomes less efficient. In this way we
can get diminishing returns by adding more labor.
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Basic Concepts, cont’d
 Isoquants, this is a form of indifference curve. (a curve that represents all the
possible efficient combinations of inputs that are needed to produce a quantity of a
particular output or combination of outputs)
 To illustrate the possible substitution of one input for another, we use
an isoquant map
 An isoquant shows those combinations of k and l that can produce a
given level of output (q0)
We are looking at different levels of the two inputs and seeing when they give us a fixed
level of outputs.
They are usually assumed to look something like this. Usually assumed to be concave to
the origin (not necessary, can think of cases where adding more gives you less utility). In
the same way as utility was increasing for the consumer, the same is usually true for the
isoquants. The more labor or capital we have (the more we move in a northeast direction
away from the origin), the higher our level of production. Anywhere on a particular line
we have a fixed level of production, q, and the axis are the different combinations of
labor and capital that can give us these production outputs. The more labor and capital
we have the higher the overall output.
Where we had RCS in the supply case we have Rate of Technical Substitution (RTS) in
this case. The idea here is in order to remain at the same level of output, as we move
down the curve how much labor has to be replaced by capital to keep the output constant.
If I reduce my labor how much more capital do I need in order to produce the same
amount? This is called the RTS. It is derived as the negative slope of the Isoquant line
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at any particular point. So this means it will varies depending on the amounts of capital
and labor and their slope on the isoquant at any particular point.
 Rate
of Technical Substitution (RTS)
 The slope of an isoquant shows the rate at which l can be substituted
for k
RTS = - Slope
It is usually assumed that RTS > 0. Assume that the tangent is downward sloping, and
because we assume that the isoquants are concave to the origin, there is usually a
diminishing effect, the slope flattens out the further you go down the curve.
How do we measure RTS? Same idea as before, the rate of change of k wrt l. That will
give us the slope of that line.
 The
rate of technical substitution (RTS) shows the rate at
which labor can be substituted for capital while holding output
constant along an isoquant
RTS  l for k   
dk
dl
q  q0
Where k is a function of l, the slope of that line is given by the derivative of k wrt l fixing
at a certain level of output.
On the next page we take this a step further (similar to the consumer analysis we saw last
class)…
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Rate of Technical Substitution, cont’d
 RTS
and Marginal Productivities
 Take the total differential of the production function:
f
f
dq   dl   dk  MPl  dl  MPk  dk
l
k
Here we have the rate of change of output wrt l (considering that q=f(k, l)). We also have
the rate of change of output wrt k. And each is multiplied by a small change in their
respective value (dl and dk).
We assume that we are keeping q constant, holding on the same isoquant. So we will set
dq equal to zero.
MPl  dl  MPk  dk  0
Now we can rearrange the terms…
MPl  dl  MPk  dk
And we arrive at …
 Along an isoquant dq = 0, so
RTS  l for k   
dk
dl

q  q0
MPl
MPk
Basic Concepts,
Different Forms of the Production Function
 Shape
of the Production Function
 The Linear Production Function
 Fixed Proportions
 Returns to Scale, if I increase both of my inputs by proportion does my
output change bu the same proportion?
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The Linear Production Function
 Suppose
that the production function is
q = f(k,l) = ak + bl
 This production function exhibits constant returns to scale
f(tk,tl) = atk + btl = t(ak + bl) = tf(k,l)
 All isoquants are straight lines
 RTS is constant
The Linear Production Function, cont’d
Notice that this production function exhibits constant returns to scale. If I increase the
amount of capital I have, say I double it, if I increase the amount of labor I have, say I
double that as well, [increasing by a constant amount] if the Production Function is
LINEAR then it will increase the production by the same amount.
For example, if I increase labor and capital each by 10% then overall production will
increase by 10%. Constant Returns to Scale. Only for this type of function and only
because it is linear. In general this will not be true. Most industries will see increased
output but as the business becomes larger the increases will fall off.
With these types of functions the Isoquants are also straight lines.
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Capital and Labor are perfect substitutes. I can get the same level of production if I
reduce my labor by adding capital in a fixed proportion. What is that fixed proportion?
It is given by the ratio of the constants b/a (the slope of the line). I can produce a
certain level of output by using all labor or all capital or combinations of the two.
Rate Technical Substitution is constant. It is the same where ever we are on the
particular isoquant.
Fixed Proportions, the other extreme.
 Suppose
that the production function is
q = min (ak,bl) a,b > 0
 Capital and labor must always be used in a fixed ratio
 the firm will always operate along a ray where k/l is constant
I can use capital or labor or ant mixture of the two. But the amount of output I get Is
limited to the minimum of these two values, q=min(ak, bl). So it’s kind of an idea that in
order to produce something I need a certain amount of capital to go with my labor. If I
add capital over and above that I get no benefit.
Fixed Proportions
If I have a fixed amount of labor then I need a certain amount of capital to produce a
certain number of units. If I increase my capital above that level I get no further benefit,
it’s just money down the drain. If I have a certain level of capital I need a certain level of
labor to produce a given number of units, increasing labor gives no additional benefit.
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The last page is telling us that we need the mix of capital and labor right at the sweet
spot, no benefit from going over.
Fixed Proportions and Linear Production are two extremes, most firms
are somewhere in-between.
The ratio of capital to labor is fixed by the ration b/a.
Returns to Scale
 How
does output respond to increases in all inputs together?
 suppose that all inputs are doubled, would output double?
 Returns
to scale have been of interest to economists since the
days of Adam Smith
We’ve already seen this from the linear production function, where we had constant
returns to scale. In that case if we double the inputs we would double the outputs. In
general this will not be the case. If I add 10% to my inputs I’m not necessarily going to
see a 10% increase in my outputs. I may see more or less return.
What will lead to increased or decreased returns to scale? Adam Smith focused on two
things? Adam Smith focused on two things:
 Division of Labor
 Specialization
Adam Smith studies people who make pin. He noticed that when they manufactured a
small order each laborer had to manufacture each part of the pin (the entire pin). As
production got higher labor could specialize, divide the job amongst themselves. As the
size of the production increased it allowed specialization and division of the labor. They
became more efficient as a result. Therefore as the size of the firm increased the firm
became more efficient. Adding additional labor actually gave more output than would be
expected by returns of scale. On the flip side however the larger you become the more
difficult it becomes to manage the operation. Therefore adding additional labor after a
certain point the firm became less efficient. It is possible to see lower returns than
returns to scale.
So it is possible for a single business to initially see greater output and greater returns to
scale and then later on as it becomes too large see lower than returns to scale. This was
Adam Smith’s discovery. This theory holds up under examination but it depends on the
type of industry you are in. If your in an industry that is very specialized then the
economies of scale last a lot longer. A good example is beer manufacturers and wine
manufacturers. It is much easier for beer to be produced in bulk then it is for wine to be
produced in bulk. It has to do with economies of scale and diseconomies of scale.
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Suppose I have the production function, it being a function of labor and capital. Then if I
increase my inputs by the same amount, say 10%, if I get the same proportional output in
my output, then I describe that as CONSTANT RETURNS TO SCALE. The linear
production function exhibits constant returns to scale. If I see less output for a given
percentage increase in the input, I increase all my inputs by 10% and I get less than 10%
increase in my outputs, we woul call this DECREASING RETURNS TO SCALE. The
opposite is the case where I double my inputs and I get MORE than double my output,
this is called INCREASING RETURNS TO SCLAE.
EXAMPLES:
Knob-Dopler
q  A(k  l1 ) where 0 <  < 1
increase k to tk and l to tl where t>0
q(tk, tl)=A(t  k  )(t1 l1 )  A(t  t1 )(k  )(l 1 )
=tAk l1  tq (k , l ).
Constant return to scale. This is an example of a function which is not linear but also
gives constant return to scale.
CES Production Function
q(k , l )  A(k r  l r )1/ r
q(tk, tl)=A(t r k r  t r l r )1/ r  tq(k , l )
This one is also constant return although it in no way looks liner.
Production Function
when q(k, l)=k  l  kl
q(tk, tl)=tk  tl  t 2 kl
if t>1, tq(k , l )  q(tk , tl )
This last example is a classic example of a non-constant return to scale. Is this one
diminishing or increasing?
This would be an increasing return to scale if t>1.
If t<1 this would be diminishing.
To consider a real world example, I open up my own store, initially I make a profit from
the one store. Say I work in Hartford, I may be able to open 4 stores and my profits grow
very quickly because I can manage those 4 stores very easily so I get increased return.
Now say I open another group of stores in Ohio, I loss the ability to manage these things
hands on, I have to hire others, another layer of management. May see the returns
diminish, I don’t get so much additional profit by adding additional stores.
In manufacturing when I am small I can specialize, I can pick the jobs I want to run. As I
become larger I have to have set mechanisms, focus on selling a certain type of output. I
can’t be as flexible as I was. Therefore, I may initially see my output increase rapidly but
later see my output decline.
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When we speak of diminishing we are not saying the output declines, it just stops
increasing at the same rate.
Returns to Scale
 Smith
identified two forces that come into operation as inputs
are doubled
1. greater division of labor and specialization of function
2. loss in efficiency because management may become more difficult given
the larger scale of the firm
 If
the production function is given by q = f(k,l) and all inputs are
multiplied by the same positive constant (t >1), then
Effect on
Output
f(tk,tl) = tf(k,l)
f(tk,tl) < tf(k,l)
f(tk,tl) > tf(k,l)
Returns to Scale
Constant
Decreasing
Increasing
same proportional output
less output for given increase
 double inputs give more than
double outputs
Returns to Scale
 It
is possible for a production function to exhibit constant
returns to scale for some levels of input usage and increasing or
decreasing returns for other levels
Economists refer to the degree of returns to scale with the implicit
notion that only a fairly narrow range of variation in input usage
and the related level of output is being considered
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Basic Concepts, cont’d
 The
elasticity of substitution () measures the proportionate
change in k/l relative to the proportionate change in the RTS
along an isoquant
%(k / l) d (k / l) RTS  ln(k / l)




%RTS
dRTS k / l  ln RTS
 The
value of  will always be positive because k/l and RTS
move in the same direction
What we are looking at here is if we change the ratio of two inputs then how does our ate
of Technical Substitution change? If I change the ratio how does it change the
substitution? This is going to give us a measure of the curvature of the isoquant.
The value of sigma is always going to be positive.
The Rate of Technical Substitution and the ratio pair are always going to move in the
same direction as long as we assume that the isoquants are concave wrt the origin. If they
are not than all bets are off.
Analysis:
% ( k / l )
d ( k / l ) RTS  ln( k / l )




%RTS
dRTS k / l
 ln RTS
We start off with a ratio of k/l, we decrease it, how does our rate of technical substitution
change? This point is greater, the rate of technical substitution is greater, as we decline
isoquant smoothes out. That’s what elasticity of substitution measures. It measures the
curvature of that isoquant line. So is sigma is high RTS is not going to change much
relative to a change in the ratio of the inputs. This means the isoquant is relatively flat.
We would be in the high levels of labor area.
If your in the high levels of capital then the isoquant is going to be sharply curved and the
sigma is going to be lower in those points. We are saying that it is certainly possible for
sigma to change on an isoquant, in fact this will almost always be the case.
(basically saying that k/l is a function of RTS, can be solved that
way then the derivative taken)
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Elasticity of Substitution, cont’d
Elasticity of Substitution, cont’d
 If
 is high, the RTS will not change much relative to k/l
 the isoquant will be relatively flat
 is low, the RTS will change by a substantial amount as k/l
changes
 If
 the isoquant will be sharply curved
is possible for  to change along an isoquant or as the scale of
production changes
 It
Optimizing Behavior
 Constrained
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Cost Minimization
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 The economic cost of any input is the payment required to keep that
input in its present employment
• The remuneration the input would receive in its best alternative employment
 Total costs for the firm are given by
total costs = C = wl + vk
Different behaviors the firm will use to optimize, minimize cost for example. Many
different ways. This example is a constrained cost minimization.
If I want to produce a certain level of output, say 1000 widgets, what is the optimal mix
of capital and labor I should put forth? What is the lowest cost to produce a fixed number
of widgets. But this is not the only question we can answer. We can answer the question
…
Constrained Cost Minimization
 Cost-Minimizing
Input Choices
 To minimize the cost of producing a given level of output, a firm
should choose a point on the isoquant at which the RTS is equal to the
ratio w/v
 It should equate the rate at which k can be traded for l in the productive
process to the rate at which they can be traded in the marketplace
Constrained output maximization. Suppose we have a cost budget, say $10,000 to spend
any way we like on a mixture of capital and labor. Given that fixed level of cost, the
question becomes, given that fixed budget how do I maximize output?
We will first look at constrained cost minimization. If I want to produce a certain level of
widgets what is the lowest cost required. Then we will look at a maximization problem
that says how many widgets do I need to produce and how do I need to manufacture them
in order to maximize my profits.
So we see there are 3 or 4 maximization problems we can look at in this area.
Total Cost of a Firm:
total costs = C = wl + vk
w: labor rate, cost per something such as employee
v: cost of capital
k: capital, cost I need to put into the business.
L: labor
Total cost is the number of units of each of the cost that I employee multiplied by the cost
per unit. (w and v are like the cost of l and k).
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So to minimize the cost of producing a given level of output how do I get to that optimal
point. We will change the mix of labor and capital so that the rate of technical
substitution, RTS, is equal to the ratio of the cost of either input. Similar to what we did
in last class. We will see that the optimal point is the mix of v and w which is tangential
to the isoquant.
Cost-Minimizing Input Choices, cont’d
Mathematically, we seek to minimize total costs given q = f(k,l)
= q0
 Setting up the Lagrangian:
L = wl + vk + [q0 - f(k,l)]
 First order conditions are
L/l = w - (f/l) = 0
L/k = v - (f/k) = 0
L/ = q0 - f(k,l) = 0

We solve this problem using Lagrangian maximization for fixed q (output to
manufacture). Cost function, price of labor times labor units plus price of capital times
capital units. Budget constraint in this case is the number of units I need to produce
minus the production function. Take the partials listed above, set them equal to 0.
Solve each of the above partials for lamda, then set the equations equal (via lamda) and
solve for W/v (ratio of the input prices):
Notation:
 w
fl
f  f
l l
 v
fk
f  f
k k
RTS=
then ...
w fl

v fk
If I can give labor and take on more capital, given the prices I have to pay for both to
make that economically sensible decision, then I will carry on doing that until the ratio of
prices in the market place are equal to the ratio of the utilization in my production
equation.
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Cost-Minimizing Input Choices, cont’d
 Dividing
the first two conditions we get
w f / l

 RTS (l for k )
v f / k
 The
cost-minimizing firm should equate the RTS for the two
inputs to the ratio of their prices
To minimize cost set the RTS equal to the ratio of market prices for the two units (capital
and labor). That will optimize capital.
We can also think about it in these terms…
Cost-Minimizing Input Choices, cont’d
 Cross-multiplying,
we get
fk
fl

v w
 For
costs to be minimized, the marginal productivity per dollar
spent should be the same for all inputs
fk
is the rate of change of my production by increasing capital, cost of capital.
v
fl
is the rate of change of my production by increasing labor, cost of labor.
w
fk
f
 l , marginal productivity per dollar spent is going to be the same. Why? Well if I
v
w
have a choice between spending a dollar on labor or a dollar on capital, and there is a
trade off between the two, I’m going to spend it on the input that gives me the biggest
return on dollar spent. I’m going to keep making that exchange until the two ratios
descend (?).
This is telling me that the return I’m getting per dollar of capital is the same as the return
I’m getting on labor per dollar of labor. If this ration equality is out of balance then I’m
going to change the mixture of labor and capital to bring it back into balance.
This is telling me if I increase my spending on capital this is the increase in output I get
given a change in capital. This is the increase in productivity per dollar of capital set
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equal to the increase in productivity per dollar of labor per dollar of labor. Therefore
I’m going to change the mix of capital and labor along an isoquant until
these two ratios are set equal. If one is higher than the other I’m going
to spend more on the higher one and less on the lower productivity one.
I’m going to change my position on the isoquant until these two ratios
are set equal.
Cost-Minimizing Input Choices, cont’d
 Note
that this equation’s inverse is also of interest
w v


fl fk
 The
Lagrangian multiplier shows how much in extra costs would
be incurred by increasing the output constraint slightly.
In this case the Lagrangian Multiplier is telling us how much extra cost I will incur by
increasing the number of units. My constraint in this problem is the number of units I
have to produce are fixed. In the maximization of utility my budget is fixed. So what
this is telling me is if I can increase the number of units produced by a fixed amount
(say one more unit) the shadow price tells us how much the costs need to increase to
produce that one extra unit. Very useful, it avoids our having to solve two
maximization problems.
This will give us an approximate idea of how much additional costs I need to produce one
additional unit (it’s only APPROXIMATE).
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Cost-Minimizing Input Choices , cont’d
We want to produce a certain number of widgets, q0. The most efficient way of my
producing the certain number of widgets is given by the level of labor and capital where
the rate of technical substitution, RTS, is equal to the ratio of prices of labor and capital.
In this case the tangent of q0 at the intersection of isoquant q0 and c2 (at the ).
So the minimum way of producing q0 is to manufacture at the c2 at the mix of l and k
indicated by the star, this will give optimal production, minimum cost. (see next page)
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Cost-Minimizing Input Choices , cont’d
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Input Demands
 Input
Demand Functions
 Cost minimization leads to a demand for capital and labor that is
contingent on the level of output being produced
 The demand for an input is a derived demand
• It is based on the level of the firm’s output
In the same way that we can construct demand functions in a utility sense, we can
construct input demand functions. Given some level of output, I can find the minimized
cost by mixing the appropriate units of capital and labor. If I want to produce so many
widgets I can do so at the lowest cost by finding the optimal level of capital and labor to
use, where the rate of technical substitution, RTS, is equal to the market ratio of prices.
The Firm’s Expansion Path
 The
firm can determine the cost-minimizing combinations of k
and l for every level of output
 If input costs remain constant for all amounts of k and l the firm
may demand, we can trace the locus of cost-minimizing choices
 Called the firm’s expansion path
If we do this, given a different level of output I can solve the problem over and over. So I
can actually construct, for a given level of output, how much labor or capital I’ll use. I
can construct, for the firm, input demand functions. Given a level of output I can tell you
the optimal level of capital needed to generate that output at the lowest cost to me and the
firm.
For a given level of output we are tracing the point which minimizes the cost. So given a
level of output, such as q3, we can tell you the level of labor and capital needed to solve
the minimization problem. (see graph below)
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The Firm’s Expansion Path
Expansion Path
 The
expansion path does not have to be a straight line
 The use of some inputs may increase faster than others as output
expands
• Depends on the shape of the isoquants
 The
expansion path does not have to be upward sloping
 If the use of an input falls as output expands, that input is an inferior
input
Two Step Process:
1) Given output level
2) Find minimum cost by finding the ratio of capital and labor I need to minimize my
cost given the level of output.
The line which shows me the optimal mix of capital and labor in the above case, we call
that he firms Expansion Path. As the firm ramps up production, the expansion path tell
me the mix of capital and labor needed at any level of production. It’s the optimal mix of
capital and labor.
How do we find it? We trace out the cost minimization choice of capital and labor.
Given a level of production I’ll find a minimized cost which will tell me the mixture of
capital and labor, that point of capital and labor is going to be the expansion path point
for that isoquant / capital/labor pair. Connecting each expansion path point gives the
expansion path. The points progressing outward represent the progressively larger
isoquant / capital-labor pairs, the northeast direction shows the firm is expanding along
these points, increasing it’s output. And this doesn’t necessarily have to be a straight line
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and it almost certainly won’t be a straight line. As I get more and more production, I may
need lessor quantities of capital relative to labor. I may need to increase the labor wrt the
point (?). So there is no reason way the path of expansion has to be a straight line. The
relative quantities of the two inputs may change. What will determine that is the shape of
the isoquants. So these will have to be very specialized curves in order for the expansion
path to be a straight line. It depends on the production and technology of the firm.
Total Cost Function
 The
total cost function shows that for any set of input costs and
for any output level, the minimum cost incurred by the firm is
C = C(v,w,q)
 As output (q) increases, total costs increase
We can also talk about the total cost function for a particular company. What will it look
like? If I have an output level, I can solve for the best mix of labor and capital I need to
minimize my cost. Therefore I can construct cost function that tells me what my cost is
going to be to produce any level of output. This will be a function of my production
function (the shape of my isoquants) and also a function of the relative prices of the input
functions.
So it’s going to be dependent on the shape of the isoquants, and also dependent on the
slope of the costs lines (k/l). So the prices of the factors, the cost of labor and the cost of
capital) are going to be important. If one factor increases it’s cost, then I am going to
have to rebalance my inputs in order to equate the RTS to the new ratio of prices. I’m
going to have to resolve this problem. If this line suddenly shifts because one of the input
cost goes up, then I’m going to have to resolve the minimization of those costs, I’m going
to have to reset the tangent of that line equal to the new ratio of prices. It is a dynamic
process.
As output increases total cost is certainly going to increase, assuming we don’t have any
negative return to any of our factors. As we focus on the cost function there are a couple
of ratios we are going to be interested in. The first one is the Average Cost Function…
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Average Cost Function
 The
average cost function (AC) is found by computing total costs
per unit of output
average cost  AC (v, w, q) 
C (v, w, q)
q
We’ve moved on now from looking at labor and capital in terms of units of input. We
are now looking at cost in total dollar terms. Given the output that we need we
can find the optimal mix of labor and capital. If I know the prices I can tell
you what the total cost is going to be to manufacture that level output. Now when we are
looking at the cost function we are truly looking at dollars worth of cost.
Average cost: if I look at the total costs of the optimal production level, of the optimal
mix of capital and labor, then I can compare total cost to produce so many units to how
many units are being produced. Can calculate the average cost in dollar terms per unit
widgets produced.
Average cost = total costs / number of units
Marginal Cost Function
 The
marginal cost function (MC) is found by computing the
change in total costs for a change in output produced
marginal cost  MC (v, w, q) 
C (v, w, q)
q
Cost is a function of input prices and units of production. Therefore to find the rate of
change of cost wrt units of output it is simply the partial derivative of cost wrt units
produced. Taking the derivative of cost with respect to quantity holding all other
variables constant and assuming the price is going to change. Consider what this will
look like graphically…
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Graphical Analysis of Total Costs
 Suppose
that k1 units of capital and l1 units of labor input are
required to produce one unit of output
C(q = 1) = vk1 + wl1
 To produce m units of output (assuming constant returns to
scale)
C(q = m) = vmk1 + wml1 = m(vk1 + wl1)
C(q = m) = m  C(q = 1)
Suppose I am going to produce 1 unit of output. To do this I need k1 units of capital and
l1 units of labor, those are the optimal mix that I need to produce 1 widget. Therefore the
cost of producing 1 widget is going to be given to me by cost of capital times the number
of units of capital plus the cost of labor times the number of units of labor. Be clear, this
level of capital and this level of labor comes from the maximization
solving problem (a few pages back). I could use different ratios of k and l to produce
the same number of outputs, but I’m interested here in producing to the optimal ratio.
Now say I need to produce m units. Assuming constant returns to scale, in order to
produce m units I’m going to use m units of labor and m units of capital, if I have
constant returns to scale. If I do have constant returns to scale my total costs is a
straight line.
Graphical Analysis of Total Costs
If I do have constant returns to scale my total costs is a straight line. As output goes up
cost goes up, there is a linear relationship between the two. What’s more, my average
cost is equal to my marginal cost. Why? Well average cost is simply the slope of this
line (since it’s straight) and marginal cost is also the slope of this line because it’s the rate
of change. In this case both AC and MC will be constant.
IF I HAVE CONSTANT RETURN TO SCALE AVERAGE COST IS EQUAL TO
MARGINAL COST
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Graphical Analysis of Total Costs, cont’d
 Suppose
instead that total costs start out as concave and then
becomes convex as output increases
 One possible explanation for this is that there is a third factor of
production that is fixed as capital and labor usage expands
 Total costs begin rising rapidly after diminishing returns set in
Initially it takes a lot of cost to produce a small number of units. There are a lot of fixed
costs say. Costs jumps at the one or two units level (small q in general). At this stage
there is not much efficiency. After that I hit a sweet spot where I add additional cost and
produce additional units. After that inefficiencies start to creep in, just become too big.
Think of it in terms of a factory, if I start off producing widgets I need a certain level of
additional factory floor, weather I’m producing one or 10 widgets.
Eventually that same labor force is able to produce 100 or even 1000 units. But if I want
to produce 10,000 units I have to put an extra shift on, I start to see inefficiencies. So
initially it is very costly per unit, then sweet spot, then after that very costly again
because inefficiencies start to creep into the system. So our curve has kind of an S shape.
Another component of this behavior may be a third factor which we are ignoring or
pushing into the background but which may be producing different effects on the cost
structure. This is plausible.
Now what’s this going to look like? Pick a point (above). Average Cost, it takes me so
many dollars to produce that many units, the average cost is the slope of the line joining
the origin and the production point. Marginal Costs is the tangential point to the slope
of the curve at the point of production.
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Consider the slope, starts steep, flattens out, becomes steep again. Initially the average
cost is going to be lower than the marginal costs, the line starts off steep. But eventually
the marginal costs is less than the average cost. It reverses. The marginal costs is going
to be greater than the average costs.
WHAT WILL THIS LOOK LIKE IF I MAP A GRAPH OF AVERAGE COSTS AND
MARGINAL COSTS?
Graphical Analysis of Total Costs, cont’d
Marginal costs starts high, diminishes, and then increases. Average costs, starts off at a
certain level, diminishes, eventually the inefficiencies give a knee up to average cost (?)
and we start to see average cost increasing. Notice if average costs are greater than
marginal costs then average costs must be falling. Visa-versa, see graph above. In fact,
if you look at the point of intersection of these two lines, that’s where marginal costs
overtake average costs.
When we talk about monopolies this point of intersection will have a huge impact on
monopolies behavior. As long as markets are not completely competitive there is going
to be an impact of marginal price versus marginal cost. That will give us the answer.
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Input Substitution
A
change in the price of an input will cause the firm to alter its
input mix
 We wish to see how k/l changes in response to a change in w/v,
while holding q constant
.
k
 
l
 w
 
v
Here we are talking about prices of the inputs, the prices w and v of capital and labor.
Suppose the ratio of the prices (w/v) change. On my maximization line the slope of the
straight line, the trade off between the cost of capital and labor, changes because the price
relative to capital and labor changes.
What we are interested in is seeing how our optimal mix of capital and labor changes
because the relative price of those two goods changes. We can see in the graph if the
ratio of the prices in capital and labor
shift then it’s going to lead us to a
different mix of capital and labor. If
we are trying to minimize cost for a
certain level of output the way we do
this is set the ratio of technical
substitution (RTS) equal to the ratio of
prices. If the ratio of prices changes
then it’s going to lead to a different mix
of capital and labor. Will lead to a
different outcome when we do the
maximization problem.
So what we are interested in is seeing
how much our mix changes given a
change in prices. So a change in the
ratio of prices, how that impacts the change in the relative portions of capital and labor.
This is what is meant by RATIO OF INPUT SUBSTITUTION.
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Input Substitution
 Putting
this in proportional terms as
(k / l) w / v  ln(k / l)
s


( w / v) k / l  ln( w / v)
gives an alternative definition of the elasticity of substitution
 In the two-input case, s must be nonnegative (due to idea that ratio of
one thing goes up you will start using the other input more).
 Large values of s indicate that firms change their input mix
significantly if input prices change (a change in prices is going to lead to a
dramatic change in the level of inputs)
w  w + ( w )
v
v
v
k  k + ( k )
l
l
l
The two ratios undergo a small percentage change. The ratio of
our input capital to labor given our initial capital to labor. This
ratio is the percentage change. So we are adding the percentage
change of each respective ratio to the initial proportion.
What we are interested in is this, if the ratio of labor cost to
capital cost increases by a small positive amount, that means
the labor costs go up relative to capital costs, by some small amount. That ratio changes
by a very small amount. Now what is going to happen to the mix of capital and labor?
Well this will lead to a change in capital and labor. Notice how these ratios work, if the
cost of labor goes up relative to the coast of capital then we will have a small positive
increase in this ratio. This is the amount of capital compared to the amount of labor.
So if this increases by a small amount you would expect capital to be substituted for
labor. Labor is becoming more expensive relative to capital. Therefore you would
think that capital should increase relative to labor (to offset the need for more
expensive labor). A small increase in this ratio should lead to a small increase in this
ratio.
So we are interested in the percentage change, so the small change in capital over labor
divided by the amount of capital over labor (the percentage change in that ratio). Divided
by the percentage change in the relative prices. If that ratio changes by 1%, what
percentage increase do we see in the mix of capital and labor that we actually use? In the
extreme this is what we get when we take it to the limit.
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Partial Elasticity of Substitution
 The
partial elasticity of substitution between two inputs (xi and
xj) with prices wi and wj is given by
sij 
( xi / x j ) w j / wi  ln( xi / x j )


( w j / wi ) xi / x j  ln( w j / wi )
 Sij
is a more flexible concept than  because it allows the firm to
alter the usage of inputs other than xi and xj when input prices
change
Here we are extending this model to look at more general setups where we have more
than two inputs. In this case the s does not necessarily have to be positive.
Short-Run, Long-Run Distinction
 In
the short run, economic actors have only limited flexibility in
their actions
 Assume that the capital input is held constant at k1 and the firm
is free to vary only its labor input
 The production function becomes
q = f(k1, l)
We’ve been talking about how businesses can change the mix of inputs given a certain
level of output that they need. But we have to think about the distinction between short
run and long run. What we mean is If we tell a company to produce a certain level of
units, if I suddenly change their target number of units, they may not be able to increase
their factor inputs in an optimal way. Some of their factor inputs may be relatively fixed
over the short term.
Example, is a factory is manufacturing widgets and they have a budget to manufacture
100,000 of these widgets, if I suddenly ask them to manufacture 120,000 widgets it may
take them a long time to increase plant equipment and capital, they may not be able to do
so immediately. They may solve the problem by using more labor. They may do that
even though that new mix of labor and capital is not efficient. The reason they do this is
because they have a certain target to hit and because they cannot vary one of their inputs
they do a “second best.” They move away from the isoquant that they would like to be
on and over the short term they are willing not to produce at the optimal level.
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Over the long run we can change our levels of capital and labor to the point where we
need them. So notice the distinction between short run and long run.
Short run, economic actors, firms have only limited flexibility in their actions. They may
find it easier to change one input where the other may be less flexible. It may be easier to
add one shift of workers than it is to add new capital to the business (such as increase the
shop floor).
So ought to assume in the short run that one of the variables is held constant. We usually
assume capital is the one held constant in short run cases and the firm is free to vary it’s
labor only. It can hire new employees and pay them overtime. In this case the
production function becomes simply a function of labor, capital remains fixed. It
becomes a single variable function. Does this change what the firm does? YES.
Because short run total costs to the firm are going to be equal to a fixed part and a
variable part. Over the short run the cost of my capital times the units of capital are
relatively fixed. Therefore my cost function becomes a fixed variable of labor and a
fixed cost of capital.
Short-Run Total Costs
 Short-run
 There
total cost for the firm is
SC = vk1 + wl
are two types of short-run costs:
 short-run fixed costs are costs associated with fixed inputs (vk1)
 short-run variable costs are costs associated with variable inputs (wl)
In the short run we usually have fixed costs and variable costs. For example, in
accounting we think of costs being fixed and variable. Most costs, over the long run, are
variable. You can always build that extra factory. Whereas over the short run, some of
the costs are constrained, they cannot be increased very easily. And other costs are
relatively variable.
Short-Run Total Costs
 Short-run
costs are not minimal costs that we need for producing
the various output levels
 The firm does not have the flexibility of input choice
 To vary its output in the short run, the firm must use non-optimal input
combinations
 The RTS will not be equal to the ratio of input prices
If we are constrained on one of our costs we are going to make a second best attempt to
produce the number of units that we need. Ideally we would vary both inputs. If one
input is fixed we would have to be very lucky to end up at the optimal level of output.
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It is more than likely that we are going to be spending more costs than we need we would
need to if we were given free long run choices to make. So we will use non-optimal
input combinations. When we do so our RTS is very unlikely to be equal to the ratio of
input prices.
Physically what does this mean? Lets suppose our capital is fixed. We can’t change the
level of capital. Given a different level of output and the fixed level of capital, there is
only one cost line which will be tangential. And that will be at that particular point. If by
some fluke we are asked to output at this level, then this is optimal (but this is unlikely).
Otherwise we are forced to use too much capital or too little capital. Flexibility comes
into the system because we can change the amount of labor that we need.
Short-Run Total Costs
Because capital is fixed at k1 we are not able to get to the optimal point on the isoquant.
We would like to be at the optimal point but if we are forced to change our production
but we cannot change our capital, then we have to make a second best choice. And we
can see that in either case we are not at the best production level. The optimal points
would have lower costs given the same level of production.
In the above graph l1 uses too much capital (k) and l3 uses too little capital. l2 is about
right but that is the point we are being driven off of due to the changed production
demand.
[End of Lecture] He will post a practice exam. Monday he will answer any practice
exam questions. Everything after midterm. Little on rational/efficient markets, and the
last three classes. The homework is a little more involved than the exam will be, less
math more interpretation.
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Optimizing Behavior
 Profit
Maximization
 A profit-maximizing firm chooses both its inputs and its outputs
with the sole goal of achieving maximum economic profits
 Seeks to maximize the difference between total revenue and total
economic costs
Profit Maximization, cont’d
 If
firms are strictly profit maximizers, they will make decisions
in a “marginal” way
 Examine the marginal profit obtainable from producing one more unit
of hiring one additional laborer
Output Choice
 Total
revenue for a firm is given by
R(q) = p(q)q
 In the production of q, certain economic costs are incurred [C(q)]
 Economic profits () are the difference between total revenue
and total costs
(q) = R(q) – C(q) = p(q)q –C(q)
Output Choice
 The
necessary condition for choosing the level of q that
maximizes profits can be found by setting the derivative of the 
function with respect to q equal to zero
d
dR dC
  '(q) 

0
dq
dq dq
dR dC

dq dq
Output Choice
 To
maximize economic profits, the firm should choose the
output for which marginal revenue is equal to marginal cost
MR 
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dR dC

 MC
dq dq
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Second-Order Conditions
 MR
= MC is only a necessary condition for profit maximization
 For sufficiency, it is also required that
d 2
dq 2

q  q*
d '(q)
0
dq q  q*
 “marginal” profit must be decreasing at the optimal level of q
Profit Maximization
Marginal Revenue
 If
a firm can sell all it wishes without having any effect on
market price, marginal revenue will be equal to price
 If a firm faces a downward-sloping demand curve, more output
can only be sold if the firm reduces the good’s price
marginal revenue  MR(q) 
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dR d [ p(q)  q]
dp

 p  q
dq
dq
dq
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Marginal Revenue
 If
a firm faces a downward-sloping demand curve, marginal
revenue will be a function of output
 If price falls as a firm increases output, marginal revenue will be
less than price
Marginal Revenue
 Suppose
that the demand curve for a sub sandwich is
q = 100 – 10p
 Solving
for price, we get
p = -q/10 + 10
 This
means that total revenue is
 Marginal
R = pq = -q2/10 + 10q
revenue will be given by
MR = dR/dq = -q/5 + 10
Profit Maximization
 To
determine the profit-maximizing output, we must know the
firm’s costs
 If subs can be produced at a constant average and marginal cost
of $4, then
MR = MC
-q/5 + 10 = 4
q = 30
Profit Maximization and Input Demand
A
firm’s output is determined by the amount of inputs it chooses
to employ
 the relationship between inputs and outputs is summarized by the
production function
A
firm’s economic profit can also be expressed as a function of
inputs
(k,l) = pq –C(q) = pf(k,l) – (vk + wl)
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Profit Maximization and Input Demand
 The
first-order conditions for a maximum are
/k = p[f/k] – v = 0
/l = p[f/l] – w = 0
 A profit-maximizing firm should hire any input up to the point at
which its marginal contribution to revenues is equal to the
marginal cost of hiring the input
Profit Maximization and Input Demand
 These
first-order conditions for profit maximization also imply
cost minimization
 they imply that RTS = w/v
Profit Maximization and Input Demand
 To
ensure a true maximum, second-order conditions require that
kk = fkk < 0
ll = fll < 0
kkll - kl2 = fkkfll – fkl2 > 0
 capital and labor must exhibit sufficiently diminishing marginal
productivities so that marginal costs rise as output expands
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